# The geometry of Nature’s stingers is universal due to stochastic mechanical wear

## TL;DR

The study shows that the common paraboloid shape of stingers, teeth, and similar structures in nature arises from mechanical wear, not just evolution.

## Contribution

The paper demonstrates that stochastic mechanical wear, not evolutionary convergence, explains the universal paraboloid tip geometry in biological and abiotic structures.

## Key findings

- Biological stingers develop a paraboloid tip shape through mechanical wear and usage, similar to abiotic structures.
- The power-law profile (z∼rn, n≈2) is a result of random erosive processes rather than evolutionary selection.
- Self-similar shapes in nonconical stingers persist due to stochastic weathering.

## Abstract

Pointed objects such as stingers, horns, and teeth have been observed to exhibit a paraboloid geometry at the tip. Interestingly, this tip geometry is not exclusive to biological structures; it is also found in abiotic forms as disparate as icicles and rock pinnacles. However, the conformity of tip shapes in biostingers has recently been selectively attributed to evolutionary convergence. In this work, we show that pointed tips of biological origin acquire the ubiquitous tip profile just as their abiotic counterparts are—by mechanical wear and usage. Our findings also explain the persistence of self-similar shapes observed in nonconical stingers, such as shark teeth and horns.

Despite their ubiquity in Nature, spikes or stingers rarely exhibit sharp tips. Instead, a closer inspection of their roughly conical tips reveals a striking similarity in their profiles: They adhere to a power-law, z∼rn, where n≈2. This conformity persists across diverse spatial scales and materials. The mechanistic basis for this universality was recently attributed to evolutionary selection for ease of piercing [H. Quan et al., Proc. Natl. Acad. Sci. U.S.A.
121, e2316320121 (2024)]. However, the transient nature of their morphology, progressively modified by repeated use and inevitable wear, has received little scrutiny. In this work, we combine tabletop experiments with continuum analysis to demonstrate that the universal tip morphology can result from stochastic weathering processes. This finding is particularly significant in light of recent observations of the same tip geometry on dissolving or melting solids and geomorphic structures in addition to biological stingers. Our results suggest that the prevalence of this power-law profile may not be the result of evolutionary selection, but rather an inevitable consequence of exposure to random erosive processes.